𝔖 Bobbio Scriptorium
✦   LIBER   ✦

A Gröbner Basis Method for Modules over Rings of Differential Operators

✍ Scribed by Toshinori Oaku; Takeshi Shimoyama

Publisher
Elsevier Science
Year
1994
Tongue
English
Weight
683 KB
Volume
18
Category
Article
ISSN
0747-7171

No coin nor oath required. For personal study only.

✦ Synopsis

We study modules over the ring (\mathcal{D}{0}) of differential operators with power series coeffcients. For (\mathcal{D}{0})-modules, we introduce a new notion of (F)-Gröbner basis and present an algorithmic method to compute it. Our method is more algebraic than that of Castro ((1986,1987)) which is based on the Weierstrass-Hironaka division theorem. The essential point of our methad consists in using a filtration of (\mathcal{D}{0}) introduced by Kashiwara (1983). This enables us to extend some of the algorithmic methods for rings of power series to (\mathcal{D}{0})-modules. As applications, we can compute, in some cases, the cliaracteristic variety, and the dimension of the space of solutions, of a system of lincar partial differential equations via (F)-Gröbner bases. The relation to previously known methods is also stated.

📜 SIMILAR VOLUMES

Gröbner Bases of Modules over Reduction
Gröbner Bases of Modules over Reduction Rings
✍ S. Stifter 📂 Article 📅 1993 🏛 Elsevier Science 🌐 English ⚖ 374 KB

Reduction rings are rings in which the Gröbner bases approach is possible, i.e., the Gröbner basis of an ideal in a reduction ring can be computed using Buchberger's algorithm. We show that one can also compute Gröbner bases of modules over reduction rings. Our approach is much more general than oth

Analogs of Gröbner Bases in Polynomial R
Analogs of Gröbner Bases in Polynomial Rings over a Ring
✍ LYN J. MILLER 📂 Article 📅 1996 🏛 Elsevier Science 🌐 English ⚖ 617 KB

In this paper we will define analogs of Gröbner bases for R-subalgebras and their ideals in a polynomial ring R[x 1 , . . . , xn] where R is a noetherian integral domain with multiplicative identity and in which we can determine ideal membership and compute syzygies. The main goal is to present and

A Generalization of Gröbner Basis Algori
A Generalization of Gröbner Basis Algorithms to Polycyclic Group Rings
✍ Klaus Madlener; Birgit Reinert 📂 Article 📅 1998 🏛 Elsevier Science 🌐 English ⚖ 656 KB

It is well-known that for the integral group ring of a polycyclic group several decision problems are decidable, in particular the ideal membership problem. In this paper we define an effective reduction relation for group rings over polycyclic groups. This reduction is based on left multiplication

Finite Length and Pure-Injective Modules
Finite Length and Pure-Injective Modules over a Ring of Differential Operators
✍ Gennadi Puninski 📂 Article 📅 2000 🏛 Elsevier Science 🌐 English ⚖ 123 KB

ww xx Let k be an algebraically closed field of characteristic zero, O O s k x , . . . , x n 1 n the ring of formal power series over k, and D D the ring of differential operators n over O O . Suppose that is a prime ideal of O O of height n y 1; i.e., A s O O r is a n n n curve. We prove that every